Theory of Nonlinear Operators: Proceedings of the fifth international summer school held at Berlin, GDR from September 19 to 23, 1977 [Reprint 2021 ed.] 9783112573921, 9783112573914

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ABHANDLUNGEN DER AKADEMIE DER W I S S E N S C H A F T E N DER DDR Abteilung Mathematik — Naturwissenschaften — Technik Jahrgang 1978 • Nr. 6N

Theory of Nonlinear Operators Proceedings of the f i f t h international summer school held at Berlin, GDR from September 19 to 23, 1977

Herausgegeben von

Professor Dr. sc. Reinhard Kluge bearbeitet von

Dr. habil. Wolfdietrich Müller

Akademie der Wissenschaften der DDR Zentralinstitut für Mathematik und Mechanik

Mit 11

Abbildungen

A K A D E M I E - V E R L A G • B E R L I N • 1978

Herausgegeben im Auftrag des Präsidenten der Akademie der Wissenschaften der DDR von Vizepräsident Prof. Dr. Heinrich Scheel

Erschienen im Akademie-Verlag, 108 Berlin, Leipziger Straße 3—4 © Akademie-Verlag Berlin 1978 Lizenznummer: 202 • 100/243/77 Gesamtherstellung: VEB Druckhaus „Maxim Gorki", 74 Altenburg Bestellnummer: 762 641 4 (2001/78/6N) • LSV 1065 Printed in GDR DDR 5 8 , - M

Preface From September 19 to 23, 1977, the fifth international summer school "Theory of Nonlinear Operators" was held in Berlin at the Central Institute of Mathematios and Mechanics of the Academy of Sciences of the GDR. I t continued the summer schools devoted to nonlinear problems organized alternately by the Czechoslovakian Academy of Sciences and by the Central Institute in Berlin. The first summer school took place at Babylon (CSSR) in 1971, the second at Neuendorf/Hiddensee (GDR) in 1972, the third a t Stara Lesna (CSSR) in 1974, the fourth a t Berlin (GDR) in 1975. The 98 participants of the fifth summer school came from Czechoslovakia, France, Italy, Poland, Romania, the U.S.S.R., the U.S.A. and the German Democratic Republic. In the 27 invited lectures and 28 communications special attention was given to qualitative aspects and, in particular, to approximation methods for the solution of various problems. It is hoped that the 52 papers contained in this volume provide an informative survey of the scientific programme of the summer school. Berlin, November 1977 Prof. Dr. sc. Reinhard Kluge Chairman of the summer school

1*

Inhaltsverzeichnis Invited lectures A. V. Balakrishnan, Boundary control of parabolic equations: L-Q-R theory . . .

11

A. Bensoussan and J.-L. Lions, On the asymptotic behaviour of the solution of quasi-variational inequalities

25

L. Bittner, Extension theorems of Hahn-Banach type and monotone operators. .

41

F. L. Chernousko, Approximation of integral functionals and numerical solution of variational problems

51

S. Fucik, Variational noncoercive nonlinear problems

61

R. Gabasov and F. M. Kirillova, On the theory of optimal control of dynamical systems

71

H. Gajewski, On Cauchy's problem for the spatially one-dimensional Zakharov system

79

K. Gròger, Evolution equations in the theory of plasticity

97

I. Hlavàcek, Dual finite element analysis for some unilateral boundary value problems 109 P. Hustàk, M. Kubicek, I. Marek and M. Marek, Bifurcation in reaction-diffusion systems 117 R. Kluge, On some parameter determination problems and quasi-variational inequalities 129 R. Kluge and H. Langmach, On the determination of some rheologic properties of mechanical media 141 J . Kolomy, Some methods for finding of eigenvalues and eigenvectors of linear and nonlinear operators 159 A. Langenbach, Bifurcation under nonlinear loading

169

H. Langmach, On the determination of functional parameters in some parabolic differential equation 175 J . Naumann, On second order evolution inequalities of Volterra type

185

J . Necas and L. Tràvnicek, Variational inequalities of elasto-plasticity with internal state variables . . 195

6

Inhaltsverzeichnis

D. Pascali, On nonlinear divergence equations

205

W. V. Petryshyn, Nonlinear eigenvalue problems and the existence of nonzero fixed points for A-proper mappings 215 S. Prößdorf und B. Silbermann, Gestörte Projekt ions verfahren und einige ihrer Anwendungen 229 K . R. Schneider, On a class of nonlinear eigenvalue problems with non-invertible linear part 239 I. Singer, Some classes of non-linear operators generalizing the metric projections onto Cebysev subspaces 245 G. Stoyan, Numerical experiments on the identification of heat conduction coefficients ! 259 G. Telschow, On an iteration procedure

269

L. v. Wolfersdorf, Necessary optimality conditions for control processes with singular integral equations and elliptic equations 279 K . Zacharias, On the numerical treatment of wave-envelope equations

293

Communications B. Brass, An obstacle problem and its solution by quasi-variational inequalities G. Bruckner, On the speed of convergence of iteration methods

. 303 309

F. Häfner and R.-J. Giesel, Numerical results of parameter identification in partial differential equations 315 E. Krauß, A general Hahn-Banach type theorem

323

Abstracts of communications H. Benker, On a problem of optimal control in abstract spaces

337

L. Berg, On the stability of discretization methods for boundary value problems . 341 J . Danes, Two remarks on functional analysis Z. Denkowski, On a relation between (K)- and (Q)-convergence of a net of sets

345 . 347

M. Fabian, Theory of Fréchet cones and nonlinear analysis

349

S. Gähler, Optimality conditions in polyoptimization

353

M. Goebel, On optimal control of elliptic systems

357

R. Hünlich, The buckling of axially compressed circular cylindrical shells with free edges 361 J . Kacur, Stabilization of solutions of abstract parabolic equations R. Kodnár, Non-linear problems of the orthogonal anisotropic shallow shells

365 . . 367

7

Inhaltsverzeichnis

E. Kozakiewicz, HeKOTopue cpaBHHTejiBHHe TeopeMH u,HajibHBix HepaBeHCTB c nocjiefleficTBHeM

RJIH

iiejiHHeñHux ftHijKjiepeH-

A. Kufner, On some type of nonlinear equations

371

375

0 . Lange, On the question of controllability of nonlinear dynamical systems . . . 377 J . Lovisek, Approximation of the Signorini problem in plane elastostatics . . . .

381

S. Meyer, An asymptotic expansion of the solution of ordinary differential equations containing a small parameter in the highest derivative 385 E. Miersemann, On eigenvalue equations in convex cones

389

K. Morgenthal, Asymptotic behaviour of some function systems

393

K. R. Schneider and B. Wegner, On the asymptotic behaviour of the period of limit cycles bifurcating from a critical point 397 W. SprôBig, Methods of function theory for the solution of non-linear partial differential equations 399 F. Unger, On the solution of a boundary control problem at the stationary heat transfer equation 403 W. Weinelt, On the numerical solution of a variational inequality

405

J . Wolska-Bochenek, On some inverse problem for a system of one-dimensional diffusion equations 409

Boundary control of parabolic equations: L-Q-R theory A. V. Balakrishnan, Los Angeles

1. Introduction We consider (finite-time-horizon) LQR problems for parabolic equations with control on the boundary (Dirichlet data). Such problems have no analogue in ordinary (i.e. involving ordinary differential equations) control and present features unique to distributed parameter systems. There is as yet no generally accepted formulation of such problems. This is not surprising considering the large variety of approaches to nonhomogeneous boundary value problems for partial differential equations. Our treatment using semigroup theory would appear to be more general than for example the work of Lions [1]. He restricts his controls to be in L2(r)112 where r is the boundary. Whatever the mathematical simplification obtained thereby, this restriction does seem physically artificial. In any event we develop a theory in which the controls are allowed to be in L2(r). Fattorini [2] is the first to treat boundary input problem using semigroup theory but he does not consider LQR problems. In a previous paper [3] we presented our first results based on the construction of a mild solution for ¿ 2 -boundary inputs. Our main objective in this paper is to obtain regularity properties of the optimal solution to the LQR problem, removing at the same some smoothness assumptions (on Q, see below) made therein. The principal result is the establishment of feedback solutions which in turns leads to a class of Riccati equations featuring unbounded, uncloseable operators. Our main tool in this extension of the previous results is to exploit (for the first time) the bound on the rate of growth of the solution at the origin — a bound we shall refer to as the Balakrishnan-Washburn bound, see below (Section 4). This enables us to go beyond the earlier theory [3] where only the analyticity of the semigroup was used. In Section 2 we develop a 'mild' solution to the boundary-control problem exploiting only the analyticity, and derive the optimal feedback solution to the boundary control problem in Section 3, in which the associated Riccati equation also is shown to have a 'mild' solution. In Section 4 we exploit the Balakrishnan-Washburn bound and show that the solution corresponding to the optimal control is actually regular, and similarly also the solution of the Riccati equation. We also indicate an example where the domain is unbounded and the boundary degenerate and yet the bound holds.

A. V. Balakrishnan

12 2. The boundary input problem

Let & denote a bounded1 domain in real Euclidean space ]Rm, with boundary J". Points in the space will be denoted f, with components Let r denote a (uniformly) strongly elliptic operator: m m g2 f m rf = EE 0 to the value at Q as n oo; or, P „ ( t ) converges strongly to P ( t ) . The limit is readily seen to be independent of the particular converging sequence Q„ chosen. Thus we see that the non-linear equation

[P(i) x, y] - [P(t) x, Ay) - [ A x , P(t) y] + [CPit) x, CP(t) y] = - [ Q x , Q y ] ,

0 < í < T < oo;

P ( 0 ) = 0,

(3.12)

where the right-hand side defines the 'forcing function', has a 'mild' solution in the sense of Browder [7]. Moreover we can show now that

u0{t) = —CP(T — t) x0(t) a.e. O ^ t ^ T

by the following argument. Define 4 * as a multiplicative (unbounded) operator on W£T) by:

A* = [/(•) € Wa{T)\ f(t) € 9{A*) A*f = g; g(t) = A*f(t) a.e.

Domain of

a.e.;

A*f(t) € TTs^)];

Then we can readily see that A * is closed with dense domain. The latter is obvious by the fact that for any /(•) in $*(/1) /(•) belongs to the domain of A * . Now from (3.3) we see that the function T

Pn(T - t) xn{t) = S*(l¡n) J S*(o - t) Q*Q„xtt(a) da,

0

(3.13)

t

belongs to the domain of A * , and converges in W^T) to the function P ( T — t) x { t ) . Further

A*Pn{T - t) xn{t) = A * S * ( l / n ) / S * ( a - t) Q*Qnxn{a) da converges to the function

i

T

A* f 8*{a - t) Q*Qx0(a) da a.e. 0 < t < T . Hence

t

P(T — t) x0(t) e @>(A*) a.e. O^i^T

and hence it follows that

M0( 0: \\AS(t)D\\^M e t- 3 l 4 - ) ] into itself and L ( T ) is moreover compact. So does L ( T ) * (defined by formula (3.9)) and is also compact. I t follows readily that I + L ( T ) * Q * Q L ( C ) has a bounded inverse on C[(0, T ) ; L 2 ( ^ ) ] and further since M { T ) a;(0) is an element therein, so is w0 given by (3.5). Hence so is x 0 . Moreover from (3.9) we can see that P ( t ) for t > 0 is compact also. Next let

z = [ I + Q L ( T ) L { T ) * Q * ] - 1 Q M ( T ) a;(0).

(4.2)

Then z is also strongly continuous on [0, T ] , by a similar argument. Next let us consider the case where z(0) is in the domain of A . Then M ( T ) a;(0) is in (^[(O, T ) ; L 2 ( $ ) ] and we shall now show that z(i) is then absolutely continuous with derivative in Lj[(0, T ) ; H r ] . First of all let

z» = [ I + Q n L { T ) * L ( T ) * Q I Y ^ Q M ( T ) x ( 0 ) or, T

z n ( t ) = Q S ( t ) x ( 0 ) - J E „ ( t , a) z n ( a ) d a ,

(4.3)

o where T

T

T

J B „ ( t , a) z n ( a ) do = Q j 8(t — a) A S ( l / n ) D J C S * ( l / n ) 0

0

a

x S * ( s — a) Q * z „ ( s ) ds d a .

(4.4)

A 2 S ( i / n ) is linear bounded, the kernel is differentiable, and hence it follows that Q t t L ( T ) L ( T ) * Ql maps W S ( T ) into ^ [ ( 0 , T ) ; L t { Q ) ) \ Hence z„(-) belongs to the latter

Since

2*

20

A. V. Balakbishnan

space also. On the other hand, we can write: T

a) zn(a) da

o t

T

t

T-t

= Q j 8(a) AS(ljn) D J CS*{i/n) S*(s - t + a) Q*z„{s) ds da t— o 0 = Qj 8(a) AS(l/n) D J C8*(i/n) S*(s + a) Q*z„(s + t) ds da. Hence T

j

T

Bn(t, a) z„(a) da — QA8(l/n) S(t)D J

- Q j A8(i/n) o

S(a) DCS*(l/n)

t

C8*(l/n)

8*(s)

Q*z„(s)ds

S*(T + a — t) Q*zn(T) da

T

+ Q J AS(\ln) 8{t - a) D f CS*(i/n) S*(s - a) Q*zn(s) ds da. 0 a

(4.5)

Hence differentiating (4.3) and collecting terms we can write finally: [I+

QnL(T) L{T)*Q*n]in

=

hn, T

Kit) = QS{t) Ax{0) - QAS(t) S(l/n) D j OS*(s + 1/n) Q*zn(s) ds o c + Q j AS(t - a + 1 /n) DGS*(T - )] into itself. Moreover both are also compact. Also the sequence of compact operators mapping £ ^ ( 0 , T); HT\ into itself: QnL(T) L(T)* Q* converges in the operator norm (over ^[(O, T); Hr]) to QL(T) L(T)* A*. We shall show now that this operator does not have (—1) for an eigenvalue. But for each n, QnL(T) L(T)* Q* has no negative eigenvalues and since the sequence converges in the uniform operator norm, the limit cannot have (—1) as an eigenvalue. Hence we have that zn = [I +

QnL(T)L(T)*Ql]-iha

and the right-side converges to [I +

QL(T)L(T)*Q*]h,

Control of parabolic equations: L-Q-R theory

21

where h is the limit of hn in ¿^(0, T); Hr]. Hence z„ converges in £¡[(0, T); Hr] and the limit has to be z since t zn(0) = Qx(0);z„(t) = / z„(a) da. o Hence z is absolutely continuous as required. Next we can make use of (3.9) to note that T

A*Pn(T)

T

x(0) = J A*S*(a) Q*z„(a) da = J j o

S*(a) Q*z„(a) da

o

= / [ £ » 0 * 2 » ] - / o o

S*(a)Q*ztt(a)da T

= S*(T) Q*zn(T) - Q*Qx(0) - J S(a)* Q*zn(a) da. o Hence it follows that A*P„{T) x(0) converges as n -> oo and since A* is closed, P„(T) x(0) belongs to the domain of A* for x(0) in the domain of A. This result being true for each T > 0, we can see that for each t > 0: P{t) x 6 @{Q*),

for

x 6 S>(A).

We can now invoke the following result which can be proved in the same manner as in the finite-dimensional case: [Pn(T) and

z ( 0 ) , s ( 0 ) ] = j \\Qxn{t)\? dt o

[.P(T) x(0), *(0)] = j\\QxSW o

dt.

Hence it follows that [Pn{t) x, y] -> [P(t) x, y]. But then for x, y on the domain of A we can now take limits in [PM

x, y] = [Pn(t) x, Ay] + [Ax, Pn(t) y] -

to obtain

[CS*(l/n) Pn(t) x, CS*(l/n) Pn(t) y] + [QS(l[n) x, QS(l/n) y]

[PB(•j fb —, and *#(m/) > 0.

(2.18)

32

A. Bensotjssan and J.-L. Lions

We then introduce za as the solution of Aza -f kz„ ^ / — z,(Aza +ccz,-f

Jt{mf),

Ì

+ Jt{mf)) = 0,

,6

TO).

J

(2.19)

By virtue of (2.18), we have (2.20)

But we have KWhhq) ^

c

(2.21)

-

Indeed, it is easy to check that ^ t. -

tuli»

where Ça is the solution of AL + «£. = / -

J?(mf),

8L 8vÀ

= 0

and from Lemma 4.1 below

from which one easily obtains (2.21). From (2.20) and (2.2), it follows that (2.22)

l|tt.llfl»(0) ^ C.

Passing to the limit, using a subsequence, we obtain that u is a solution of (1.12). By the uniqueness of the limit, we obtain the first part of the desired result. Let us now assume that JK(mf) < 0.

(2.23)

We note that we can write (1.5) as Au„ + otu, = / — g„,

g„ ^ 0,

(2.24)

and (2.25)

l&Lvg» Ss 0. With the definition of wa and

we can rewrite (2.24) as follows

Aw, + «», + !. = / - g„.

(2.26)

We can extract a subsequence such that w, —> w in H2{(9) weakly, K ga ->g in L2(0) weakly.

(2.27)

Asymptotic behaviour of the solution of quasi-variational inequalities

33

From (2.26) we obtain, Aw

+ A = / -

fall dw — dvA

g,

= 0,

weH^O),

g ^ 0.

(2.28) (2.29)

We also havefir«(aw« + h ) = 0, therefore Xg = 0.

(2.30)

If A = 0, then from (2.28) we deduce that J?(mg)

(2.31)

= Jt(mf),

which is a contradiction, since the left hand side of (2.31) is 0, and the right hand side of (2.31) is strictly negative. Therefore, we must have g = 0, and Aw

+ A = /,

A -

JSf(mf),

hence

— = 0,

8Va r

and the second part of the theorem is proved, under the assumption (2.23). We finally consider the case when = 0.

Jf{mf)

(2.32)

We turn back to (2.28) which is still valid. We have A =

—Jtiyng),

hence from (2.30)

gJi{mg)

= 0,

which necessarily implies g = 0, A = 0, hence again the desired result. I t remains to show that if (2.32) holds, then converges. But turning back to (2.19), we can see that in the case (2.32) ua = za and thus ua remains in a bounded subset of H2( u. From (2.24), we obtain, passing to the limit, Au = f-g,

g^ 0.

(2.33)

From (2.33), it follows that Ji{rnj)

= Ji{mg)

=

0,

hence g = 0. Therefore u satisfies Au = /,

u ^ 0.

(2.34)

The solution of (2.34) is not unique. Let us prove that it has a maximum element. We shall check that w« ^ u 3 Kluge

(2.35)

34

A. Bensoussan and J.-L. Lions

for any solution of (2.34). From (2.3) and (2.34), we deduce A{ul - «) + ), 8vÂ

f, (4.10)

We multiply (4.10) by m, and write it as ^ 8 ( 8u°\ mai - E — 8x{ \ > 8 x j J + txuejn

e

^ 8m 8ue a 8xi ->87 xT j +

m ( u e s ) + + m H ( x , VueJ

8u° a „ ^ m •8xì

= fm.

(4.11)

37

Asymptotic behaviour of the solution of quasi-variational inequalities

We multiply (4.11) by

(««)+

"p-i

and integrate over 0. Taking into account the

equation for m, we obtain dxj

+

m[{ul)+Y

dx + j

0 =

/m

/

f>;>

-]p-2 [(m^)+]2 dx

Sx,

mH{x,

Vu*)

-

K)+

e

p-i

dx

'0 iK)+

["«

^

and using (4.1), it follows that V

+

gp-i

lS

|V(m®)+|2 [ K )

m(j

+

? -

2

dx

0

j

1

m[(ulYf

[(u'XY

*d x - ^ J m

e> £

•I

|V(tt')+|

(«;)+

(4.15)

< CC0[l + to

and thus if CC0k0 < 1, (which is the condition k0 sufficiently small), we obtain from (4.15) that llM,«ll(F!.i>(0) = const

(independent of oc, e).

(4.16)

Since p > n, it follows from (4.16) that Kll c . ( 0> ^ const,

^ const.

(4.17)

Letting s tend to 0, we obtain RIU»

C,

\\wa\\cam < C,

IIVw.llc.,0

C,

\oc.Jf(mua)\ < C.

(4.18)

W e now use an argument of J. M. Lasry [4]. Let x0 be a fixed point of 0. W e consider two mutually exclusive cases. Either ua(x0) —00, or it has finite accumulation points. Let us first consider the case when ua(x0) - > —00. Since ||VMa||co(^, is bounded, it follows that Hwallco^) —00. Hence for oc sufficiently small, we deduce from (4.2) = 0.

AuB + H(x, Vua) +au,-f

(4.19)

Setting w, = wa — ||w>a||C0(p), we can also write (4.19) as Aw, + H(x, Vwa) + ¡xua — f = 0 and

^ 0,

dvA r

=

0.

W e can extract a subsequence such that w«

w in W''r(6l)

weakly,

X, and we have tin dvA1 r Aw + H{x,Vw)

+ X-

and A, w is a solution of (4.4).

= 0,

f = 0,

weW''p((P)

39

Asymptotic behaviour of the solution of quasi-variational inequalities

In the second case, there exists a subsequence such that ua(x„) converges, therefore for this subsequence, IIwJIh«.» remains bounded. By extracting a new subsequence, we see that ua u, where u satisfies

Au + H(x, Vu) ^ /,

u^

u[Au + H(x, Vu) -f]=0,

0,

—= 8va

u e w ,- r(&)

0,

and w = u, X = 0 is a solution of (4.4). The proof of Theorem 4.1 is complete. 4.3. The case of equations

In the case of equations, we do not need that

f, h belong

to

Lf{(9).

We assume that

8H k , sufficiently small, H is measurable, 0

(4.20)

h(x) = H(x, 0) € L ( K for each i £ I. Let the given linear mapping f0: L0-> K now satisfy the constraints f0{x) < pi{x)

\/x € L0 n Ci,

ie

I.

This problem with K = R and the result below is due to Landsberg and Schirotzek [7] apart from the way to deduce it. Lemma 3. a. Necessary for the existence of an extension f1 for /„ is condition (C): For each finite subset I0 a I, corresponding X{ 6 C\ and nonnegative numbers A; (i £ /„) with 27 kxi £ L0 the following inequality holds i€/0 \ie/0

/

ie/o

b) Sufficient for the existence of an extension / is condition (C) in connection with condition (D)

uC.-cLo-co/uCA. ia \ai )

P r o o f , a) Follows immediately from /(x;) Pi(%i) and / = / 0 over L0. b) I n the sequel I0, J0 denotes an arbitrary finite subset of I and m;, w; any natural number. Evidently C:= co / U CA = I £ L \ia I lie/. ; = 1

I h 0, xif € cX J

€ Cij,

since C; is convex, hence y[..

m


0). Now define mt

{

%i =

A,-,-, x{ = 27 Y a;,-,- 6 C

mi

1

27 727 ie/„ = 1 kjPi{xij)

| x = »€/. 27 j27 > 1, > o,«i?- € c AJ = l ^ ¡ j - Jo xn w i t h ¿a ^ 0, xn eCi (j = 1 , . . . , m - i e i0), and take account of the fact that condition (D) is equivalent to the condition C 1

i.e. a linear mapping f-.L^-K

czL0—C.

satisfying f(x) = f0(x) V x € L0 and f(x) iS pi(x) V x e Cit i e I.

Extension theorems of Hahn-Banach type

45

As a result there is a w 6 L0 and y E £ PiiHij € C with ¡i^ >_ 0, 6 C,- (j = 1,.. ; i d J0 cz I) such that x = w — y. Without loss of generality we may temporarily suppose that 7 0 = J0, mi = a;,-,- = (we can achieve this by adjoining ¡x^ = 0 or Xij = 0 properly chosen) and that each sum E (kj + t*ij) over j is > 0. Then

where

mi x + y = E E {kj + /¿a) xn = £ ¿e/o j=i ie/o {hi + Pi}) > o,

>

xi=z

' T

^

€

•

Hence condition (C) and the convexity of pi imply /.M

= fo(x + y)^£*iPi(xi)

t i £ k £

nt mi /o(w) - £ £ ¡¿ijPiiya) ^ £ £ i£J0 j=1 ie/o j=l

Aii

| ^

Pi{x>j),

kjvMn)-

Let us take into consideration all representations £ £ kj x ij for x, while x, y and the representation E £ ¡¿iiVij f° r y being fixed. Then the last inequality shows that the right-hand side is bounded below, thus its infimum, i.e. p{x), is well defined. That p is positively homogeneous, i.e. p(kx) = Xp(x) \/x (L C and X 2: 0, is obvious. p is also subadditive, since for aj 6 C, y £ C there exist representations x = £ £ ¿¡Fa,

y = £ £ Pay a •

Without loss of generality we may again temporarily assume tation for x + y reads

= y^. Then a represen-

x + y = £ £ (hi + pa) xif. According to the definition of p we have p(* + y ) ^ £ £

+

Pi(xa) = EEkiPi(xa)

+EEnnPi(yn)-

Fixing the representation for x, but varying that for y, we obtain at first p(* + y) ^ E E hiPi(xa) + p(y) and then p(x + j/) fS p(x) + p(y). L0, /0, C, p fulfil the assumptions of Lemma 1. For let x be in C n L0, mt x = E E kjXij, •e/o i=i

ki

o,

Xij € Gi

be any representation of x as element of C and k

=

E kj > j

xi = E

j Ai

xij,

46

L . BITTNER

then due to condition (C) of Lemma 3 /«(*) = MZkxi)

^ E kPiixi) = EE

W n ) .

hence f0(x) sS p(x). Therefore Lemma 1 applies. A linear mapping / : L -> K exists such that f(x) = f0(x)

V* £

/(») ^ P(x)

WxeC

= co({J C-).

If a; £ Gi, then £>(a;) Pi{x) according to the definition of p, consequently f(x) 5S Pi{x). Thus / is an extension for / 0 .

3. Monotone operators and variational inequalities Let A be a set-valued mapping from a subset D{A) of a real (topological) vector space Y into a certain linear set X = L0(Y, K) of linear (continuous) mappings: Y K, where K denotes a real, partially ordered, conditionally complete (topological) vector space. Let {x, y) denote the value of a; € A>( Y, K) at y £ Y. By a monotone operator we understand a mapping satisfying {xx — x2, y^ — y2) S: 0 V Xj £ Ay,, yf £ D(A) (j = 1, 2). As a rule the set R of real numbers is taken as space K. Let us suppose that there is given a family of usual monotone operators Ait i £ I, (i.e. Aiy £ L(Y, R)) with common domain D(Ai) = : D(A), then the conventions K := R1 with the componentwise ordering and linearization, ((« 0

(1.2)

for any arbitrary nonzero vector f = (£1; £2, f 3 ). Condition (1.2) guarantees strict convexity of the functional J(u). We shall construct difference approximations of (1.1). For this we partition the domain D into elementary cells, replace integrals over the cells by approximate quadrature formulas, and approximate the value of the function u and its derivatives in each cell by the value of the function at the grid points (vertices of the cell). The total error of the approximation of the functional will be conditioned by the approximate character of these three operations. In what follows we assume that the partition of the domain D into cells fits boundaries of D and that this operation is exact (it does not contribute to the total error). This condition can often be satisfied by means of mapping of D on a rectangular domain. For uniqueness of the solution of the problem of minimizing the functional and for the convergence of the computational algorithms we require that the approximation scheme be a strictly convex function of the values of u at the vertices of the cells. The construction of strictly convex schemes, as we show below, is achieved by increasing in the quadrature formula the number N (N ^ 1) of times that the function / in the integral is calculated in each cell. We shall confine ourselves to such approximation schemes that for each cell use only the values of u at the vertices of this cell. Such approximations are convenient for methods of minimization, especially for the method of local variations, because a variation of u at one point changes here only several terms of the integral sum. For each cell we shall define the relative error x of the approximation scheme to be the ratio of the computation of the integral over a cell to the area of the cell. The error in the computation of the integral over the entire domain D can be estimated as the maximum over the cells of these values. Denote by e the maximal linear dimension of a cell. The maximal order with respect to s as e 0 of the relative error x does not exceed e2 for an arbitrary number jV Si 1 of applications to /, i.e. x = 0(e 2 ). I n fact, for any N, knowing u only a t the corners of a cell, we can obtain accuracy of order not greater than ea in the approximation of / in a cell, and the integral over a cell can be computed with accuracy not greater than e4. Since the area of a cell is equal to e2, the order of the relative error amounts to » = 0(e2). For given accuracy of approximation it is expedient to choose schemes using the least number N of applications to the computation of the function in the integral, i.e. the most economical schemes.

Approximation of integral functionals

53

Taking the requirements stated above into account, we will aim at constructing schemes satisfying the following conditions: 1) for each cell the scheme uses only the values of u at the vertices of the cell; 2) the order of the relative error is k = 0(e 2 ); 3) the scheme preserves strict convexity; 4) the number N of times per cell that / is computed is minimal. 2. Let us first investigate the partition of the domain D into equal parallelograms. Without loss of generality, in this case we can confine ourselves to the consideration of rectangular cells; the corresponding transformation of the cells is achieved by affine mapping of the independent variables. Let us denote the lengths of the cell sides by Ax,

Ay.

Consider a single cell Qij of D with corners at the points (xi, yj), {xi+1, yj), (xi, yj+1), yj+1), where = i Ax, yj = j Ay, i = 0, 1, . . . , j = 0, 1 , . . . , and let the values of the grid function at the respective corners be u°, u1, u2, us. Write the integral J as the sum of integrals over the cells Qij, i.e. J = Jij• To calculate the integral E

\

m=0

3 OS njftl m>

E

m=0

AX

3

E m=Q

Ay

>

/

(1-3)

where cs, Xs, fis, y„ are parameters of the scheme. Condition 1) is satisfied here. I t can be shown [6] that in order to satisfy condition 2) it is necessary and sufficient for the parameters of the scheme to satisfy the relations: ¿ c

3=1

s

=

1,

5= 1

/*S + 0i = y! + y5 = o> E

+

c (*l

8 = 1s

«5) =

=

5=1

E

¿


I*, I' —1* when h 0) with estimates 0 ^ /' - /* ^ Kbhe~3,

0 ^ P - I* ^ K6he~2

(2.7)

(when u' = u°). The p r o o f of Theorem 2 is based on Theorem 2.2 of the book [2] and is given in [6]. The constants Kx, ..., K6 in (2.5) —(2.7) can be calculated explicitly for given function / and domain D (see an example in [5, 6]). 2. Now we shall consider the convergence of the exact solution (u*, I*) of the approximating problem to the solution (u, J) of the initial variational problem when e 0 in (2.4). This question was studied in a number of papers, see, for instance, [7—10]. We shall give here a theorem obtained recently by A. A. Lyubushin for the variational problem (1.1) with constraints (2.1). The following assumptions are made. The domain D is a rectangle, the approximation scheme (1.12) for triangular grid is used. The functions u+, u~ in (2.1) are twice continuously differentiable in D. The function / in (1.1) is twice continuously differentiable and satisfies condition (2.3) in a domain G defined by (x, y) e D,

—oo uy

u-(x, y) — 6 iS u

< oo,

) [ d > 0, J

u+(x, y) + 6,

(2-8)

where H is a nonlinear perturbation. 1.2. The operator L. Let H be a real separable Hilbert space with the inner product (u, v) and with the norm ||w|| = (u, w)1/2. Suppose that B:H -> H is a linear completely continuous selfadjoint operator and let A be the sequence of all eigenvalues of B calculated together with their multiplicity. Let £ H, ||ej| = 1, be the normalized eigenvector of B corresponding to A € A, i.e. = Bex, Choose the eigenvalue

A {w), x : r h > inf

JEM

are bounded. Let fdL2(Q), cp € Gk-\Q) n W"-\Q). Define the functional J g[WMu(x) + ^M 0, r0 > 0 such that m dp £ — (;c, y,

0

(19)

i=i 8y{ for \y\ — c0, \x\ if' r 0 and \z\ ^ c0 if m> 8F (x, y, E1, —

0;

0

(20)

¡ = 1 0Z; for \x\ ^ c0, |2/| ^ c0, and |z| = c

0

i f q > 0;

y, z) ^ -F(0, y*, 0) for \z\ g c0, \y\ g c0, |j/*| ^ c0 and \x\ = r 0 . T/i-ew fftere ermi-s (cc0, j/0, z0) € R"+ m + ? with |i/ 0 |^c 0 , [z0l^c0 and

(21) (22)

VF(x0, J/«, z0) = 0.

(23)

S k e t c h of the proof. a) It suffices to prove the assertion under the slightly stronger assumptions that in (19) —(21) the strict inequalities hold and that F is twice continuously differentiate. The general situation can be proved by approximating the function F by a uniformly convergent sequence of functions satisfying these additional assumptions and by the limiting procedure. b) If F satisfies these stronger assumptions and m = 0 then the maximum of F(x,0,z) on the set \(x, z) (E IR"+S; \x\ r0, \z\ ^ c0) must be attained at an interior point (x 0 , z0), so VF(x0, 0, z0) = 0. c) Suppose m > 0. If F satisfies the stronger assumptions from a) then we modify F outside of an open set containing a = {(x,y,z)

6R«h-w;

\x\ ^ r0,

\y\ ^ c 0 ,

|z| ^ c0}

such that VF is bounded on IRn+m+i. Let • H satisfying the growth condition ||S«||

+0IMI4,

uzH,

where a 2: 0, /? ^ 0, S € [0, 1]. In the above mentioned papers there are given the connections of these results to those previously obtained by various authors. 5.2. The additional assumptions (31), (33) in Theorems 4.2 and 4.3 are important for the used method, and the variational proof of Theorem 1.5 without these assumptions is an open problem up to now. Note that if X0 = max/1 then the additional assumptions upon 8 are not necessary (in this case Z = {0}). 5.3. The characterizations (29) and (30) of the solutions of (1) can be useful for using numerical methods for construction of the solution of (1). Unfortunately we had no success to obtain that the solutions of (1) can be characterized as the saddle points of the functional 0 .

References [1] S. Ahmad, A. C. Lazer, J . L. Paul, Elementary critical point principle and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J . 26 (1976), 933—944. [2] S. Fuöik, Nonlinear equations with noninvertible linear part. Czechoslovak Math. J . 24 (1974), 467-495. [3] S. Fuöik, J . Neöas, V. Souöek, Variationsrechnung. Teubner Texte zur Mathematik. Teubner, Leipzig 1977.

Variational noncoercive nonlinear problems

69

[4] S. Fufiik, Nonlinear potential equations with linear parts at resonance. Casopis Pest. Mat. (to appear). [5] S. Fufiik, Nonlinear equations with linear part at resonance. Variational approach. Comment. Math. Univ. Carolinae (to appear). [6] S. Fu6ik, Ranges of nonlinear operators. Lectures Notes of Charles University. Charles University, Prague 1977. [7] R. E. Gaines, J . L. Mawhin, Coincidence degree, and nonlinear differential equations. Lecture Notes in Mathematics, Vol. 568. Springer-Verlag, Berlin 1977. [8] E. M. Landesman, A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance. J . Math. Mech. 19 (1970), 609—623. [9] A. C. Lazer, Some resonance problems for elliptic boundary value problems. In: Lecture Notes in Pure and Applied Mathematics, Vol. 19: Nonlinear Functional Analysis and Differential Equations (ed.: L. Cesari, R. Kannan, J . D. Schuur), 269—289. Marcel Dekker Inc., New York, and Basel, 1976. [10] A. Kufner, O. John, S. FuCik, Function Spaces. Academia, Prague and P. Nordhoof, Groningen, 1977. Address

of the author:

Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovski 83 186 00 Praha Czechoslovakia

On the theory of optimal control of dynamical systems R. Gabasov, F. M. Kirillova, Minsk

This paper is devoted to three sections of optimal control theory: singular optimal controls, optimization of non-linear systems with delay and methods of functional analysis [1 — 5], Results of the authors and their collaborators are presented, essentially.

1. Singular optimal controls Consider on the class of piecewise continuous r-dimensional functions u(t), t0 assuming values on a given set U,

»(f) € U,

t Si tls

(1)

the problem of minimizing the functional I(u) = ?(*&))

(2)

determined on the trajectories of the system

x(t) = f{x, u, t), x{t0)=xo, a; = {xx,..., xn\.

(3)

The moments t0, tx and the vector x0 will be assumed to be fixed. As to the functions (p(x), f(x, u, t), we suppose that they are continuous together with 8°(t), u°(t), t) = max H(x°{t), y>°{t), u, t). uOJ H{x, y>, u, t) = ip'f{x, u, t), .o= _ eff(af>(t), y0, u°(t), t) 8x y°(h) = -

[

8x

a

'•

(6)

72

R . GABASOV a n d F . M . K I R I L L O V A

Calculating from (4) the control u = u(x, y>, t) and inserting the result into (3), (5), we obtain a boundary value problem of 2n equations for the computation of the optimal trajectory. This procedure is realizable if the maximum in (4) is attained at a single point. It may, however, happen that this maximum is attained at several points or that the function H does not depend on the parameters u. Such cases are called singular and controls along which they are realized are called singular controls. In other words, singular controls are such controls for which the maximum principle degenerates and thus becomes noneffective. Presently a great deal of applied problems are known where singular controls are discovered. One can say that the singularity condition, in a sense, is a signal that the problem is complicated enough [1] and that it cannot be studied completely using only the Pontryagin maximum principle or another necessary first order optimality condition. The problem of singular controls has been investigated rigorously for about 15 years. In 1963—1970 in the arsenal of means of the theory of optimal processes there was involved a series of methods substantially differing from the method by which the Pontryagin maximum principle had been proved. As a result there were obtained the necessary optimality conditions of Kelley, Robbins and KoppMoyer [6], effectively working with singular controls from open domains. The possibilities of investigating singular regimes are substantially expanded by using variation bundles introduced first in [7]. Definition. The control u{t) is called singular in problem (1)—(3), if for each t, to Sa t sj tu there exists a subset a> cz U such that H(x(t), V(i), u, 0, be optimal in the problem (1)—(3) it is necessary that 1) the maximum condition (4) holds on T \ a, 2) the inequality Avf'{x»{t), u°{t), t) W°{t) Avf{x°(t), u°(t), t) + r o'(i)

8Avflx°(t), u°{t), t) 1 Avf{x»(t), «»(I), t) < 0 ^ 7 8x

holds for all v £ to, t 6 a {see [1]).

74

R. Gabasov and F. M. Kirillova

With the development of optimal control theory the interest in high order optimality conditions increases. The conditions for optimality of singular controls, decsribed above, are the first results of the theory of necessary high (second) order conditions for optimality. In the general case the problem of finding auxiliary optimality conditions for controls satisfying the first oder conditions is extremely complicated [1, 14—16]. Recently there appeared papers on necessary conditions for optimality in problems with boundary conditions [17—19]. 2. Necessary conditions for optimality in systems with delay In the theory of optimal processes systems with delay appeared soon after Pontryagin's maximum principle had been proved. G. L. Kharatishvili extended the technique of proving the maximum principle to a system with constant delay. Further investigations of problems of optimization in systems with variable and distributed delay showed that the conjugate equations, described with the aid of the function H(x, ip, u, t), have a very complicated structure and are of little use for practical computations. One can obtain a law which is invariant with respect to a number of classes of control systems and which controls the conditions for optimality and the conjugate equations if instead of the Hamiltonian H a special functional is introduced. Consider the system

(10)

^=X(x(t,.),u(t,.),t), where

= {®i(0. t0-h^t^t0],

{02(t), t0-h^t^t0], x(t, •) = {^(r), ..., x„(r), t — h^z'(t) X(x(t, •), u(t, •), i) dt. to If u(t), u(t) = u(t) + Au(t), t 6 T, are admissible controls, x{t), x(t) dcp(x(ti)) sponding trajectories, then assuming iptfx) = we obtain 8x AI(u) = — [ji(x, xp, u) — n(x, y>, u)]+rj, «i

dn{x, %p, u) dn(x, Ax(t) y, u) dt dx(t) Sx(t) i. h + 0(11^)11) + / ox^\Ax{t, .)||) dt.

V=-f

are the corre-

(11)

75

Optimal control of dynamical systems

3tz Here — is the classical variational derivative of n{x, y>, u) with respect to x(t) (first dx kind variational derivative). Suppose that there exists e0 < + 0 0 such that for all e, 0 < e Si e0, on the needle variation j « ( o = 1Ir0 '' [v, the relations

0 t€[0,6'

i?[e

||zla:(i)|| ^ Ke,

/

ôx(t)

+

fi)'6 + e),

+

ve

Ü, '

e e T

K = const,

^ ~

8n{x:Z:

ôx(t)

(12)

j| « ^

Kl

=

C O nst,

(i3)

hold. We define the function a(v, 8) by presenting the increment n(x, ip, u) — n{x, y>, u) in the form n(x, f , u) — n(x, y), u) = ea(v, d) + o(e) S 7t(iC V) ilS and call it the second kind variational derivative — — = a(v, t). 8u(t) From (11), on the basis of (12), (13), we obtain the following necessary condition for optimality in the problem (10), (2). Theorem 3. Let u°(t), t C T, be an optimal control, x°(t), ip°{t) the corresponding solutions of the equations x(t)=

dn{x, y>, u) , dtp

Sn{x, y>, u) r-rr—> ox(t)

W) =

te(t0,h)-

Then the maximum condition ¿«(0

-

»

»

>

holds. The inequalities (12), (13), which characterize the continuity of the solutions with respect to the parameter and to perturbations which are small in the average, are essential conditions in the problem under discussion. One can show that the maximum condition (14) can be violated if the inequalities (12), (13) are not satisfied. Other necessary conditions for optimality in terms of the functionals n(x, ip, u) are given in [2]. From the classes of systems for which one can use the above described method of solution we note control systems with a high order differential operator and systems of integro-differential equations [2].

76

R . GABASOV a n d P . M . KIRILLOVA

3. Methods of functional analysis in the theory of linear optimal systems The first method of functional analysis explicitly employed for the solution of problems of optimal control was based on the theorem of separability of convex sets [20]. I t was applied to the solution of the time optimal problem in linear systems by R . Bellman, I. Glicksberg and O. Gross [20]. The same theorem was used by R . V. Gamkrelidze [21], who was the first to prove that the time optimal control in the linear system — = Ax + bu, dt

(15)

where \u(t)\ sS 1, has the form u°{t) = sign y>°'{t) b, ^

at

(16)

= -¿VW

(17)

(which is equivalent to the maximum principle). N. N. Krasovskiy's method of moments appeared to be the second method of functional analysis which had a great influence on the development of the theory of linear optimal systems. For studying the time optimal problem N. N. Krasovskiy used results of M. G. Krein concerning the ¿-problem of moments. The application of the Z-problem allowed to reduce the variational problem to the calculation of a finite-dimensional vector in a minimum problem for a convex function. In other words, it was shown that the initial condition ip(t0) for system (15) with the aid of the construction of the L- problem can be found as a solution of some finite-dimensional problem [22], The analysis of the method of moments shows that it is applicable only to the investigation of systems of the form ^

= A(t) x(t) + B(t) u(t)

with restrictions of the type |[w|| L. In the paper [23] there was proposed a general scheme of investigation of linear optimization problems based on the use of the separation theorem for convex sets. An important feature of the new method was that the class of problems solvable by this method substantially expanded. It became possible to study systems of the form dx — =A{t)x dt

+ b(u, t),

ueU,

(18)

where U is not necessarily a convex set, b(u, t) is not necessarily a linear function. As well as in the N. N. Krasovskiy method the result contained the maximum principle, and the determination of the vector ip(t0) was reduced to the solution of a finite-dimensional optimization problem. Consider the problem of minimizing the functional (2) on the trajectories of the system (18).

Optimal control of dynamical systems

77

The idea of the method [23] is the following. Consider the set of attainability of system (18). Let i2((p* M 2 + |«*|2) + |«„|») dxJ. We identify the spaces H and H„ with their duals H* and H* and denote the pairing between V* and V by (.,.) and between F* and V„ by (.,.)„. We shall frequently use the following simple relations IMIc. ^ ^2n IMIn + 21|«||. I Win,

(1.2)

INI2C ^ 2 M | | « L ,

V « e F,

(1.3)

p(n) |M(W)|^c|||W|||2n,

V w € F p „,

(1.4)

p»(»)|«(»)|»^c|||«|||»,,

VM 6 F 2 „,

(1.5)

\/ueVp,

(1.6)

p(x) |w(a;)|2

as |a;|

oo,

i>3(x) |M(a;)|2->0 as |z| ->oo, 6 Kluge

vuevn,

V«£F|.

(1.7)

82

H . GAJEWSKI

Here and in what follows c, cu c 2 , . . . denote constants independent on n whose exact amounts are irrelevant. L e m m a 1.1. The linear extension operators defined by

^f \x\ > (Fl„u)

(x) =

(F\u)(x)

u(x)

if

n,

u{n) e x p (-p2n(\x\ — n)2)

if

\x\ > n, pn =

p(n),

if |x| = {w(a;) =n> \ (u(n) + ux(n) sgn x{\x\ — n)) e x p ( — p\{\x\ — n)2) if x >

pn

F°pn € (Hpn -> Hp), respect to n.

|

n,

Flpn £ (Vpn - > F p ) ,

€ (F£„

F 2 ) are uniformly

bounded

n

with

P r o o f . Evidently, Fpn is uniformly bounded. In order to show the uniform boundedness of Fpn and Fpn we need the estimates p(n + x) ^ pn + p0 \x\

VxeR1,

(1.8)

00

J y" exp ( - 2 / ) dy ^ c, o

i=l,...,6.

(1.9)

Now, using (1.4) we find for u € Vp„ oo l l l ^ l l l *

=

I I N I l L

+

2 \u(n)\* / n

(?»(*)

+

4pi(x -

n f ) exp ( - 2 p \ ( x -

n f )

dx

oo =

\\\u\\\l„ + 2 \u(n)\*j (p*

+

+ 4p2nyj exp (-2 + 2 / {p*{\u[n)\ + \ux{n)\

-

™))2 + P 2 [ K W I + 2 2 > 2 (x -

n

X (|«(»)| + \ux(n)\ (x - rc))]2 + [2rf(|»(n)| + 2 \ux{n)\ (x - n)) + 2p2n(x -

n) (\ux(n)\ + 2p2tt(x -

w)) (|»(»)| + \ux{n)\

-

»!

X exp (—2p 2 (z — nf) dx ^IIMIlM 1 +

±fiL+Pai.YIL+y)'+L+p»±Y

PnJ p\ 0

i\

Pn)

\Pn

Pnj

\

Pn)

n)

Cauchy's problem for the one-dimensional Zakharov system

x / i +pny

( - +

\

\Pn

X |l +pny

+ - )

\PI Pnll

L

8S

\Pn

+p«y

P»J

+ ^ - j j 2 J exp ( - 2 » « ) d ^ j ^

Cl

|||«|||r:.

Remark 1.2. Proving Lemma 1.1 we did not use that p(x) - > oo as \x\ quently, Lemma 1.1 holds also for p(x) = 1.

oo. Conse-

Remark 1.3. In what follows we shall identify functions defined on [—n, n] with their extensions on R 1 given by Lemma 1.1. Accordingly, we can consider the spaces H„,..., Vp„ as continuously imbedded into the spaces H, ..., respectively. Throughout this paper let S = [0, T] be a bounded time interval. I t is convenient to introduce spaces of functions on 8 with values in various Banach spaces. For a Banach space B we denote by — C(S; B) the Banach space of continuous functions provided with the norm INIc(s;s) = max ||m(

I M I i o o ^ n *

IIm2|Ic

Theorem 3. Suppose ua £ V2 and va, wa £ V. Then the problem (2.1) —(2.4) has a unique solution (u, v, w) vrith the following properties uiCw{S-,Vl)nC{S-,Vp), v, w £ CW(S; V) n C(S; H),

(2.7) ut, vt, wt 6 C^S; H) n C(S; V*).

(2.8)

Theorem 4. Suppose ua £ V2 and va, wa £ V. Then the problem (2.1) —(2.4) has a unique solution (u, v, w) satisfying (2.8) and u £ Cw(S • V2) n C(S; V).

(2.9)

Cauchy's problem for the one-dimensional Zakharov system

85

Remark 2.2. The solution (u, v, w) guaranteed by Theorem 4 satisfies because of (2.8), (2.9) and (1.3) the inequality max max ([«(i, x)\ + |v(t, x)\ -f- \w(t,

< oo.

tts . xeR 1

Consequently, a collapse as defined in [14] can be excluded. Theorem 5. Let (e„) he a sequence a sequence with

uan e VI,

of non-negative

van = van,

numbers

and let ((wa„> va„> waJ)

wan = wan 6 Vn.

Then, for n — 1, 2, . . . , the iunt + unxx

'problem

— w„u„ = 0,

v nt + K

+

w„t+vnx

= 0,

Kl2 -

e„vnx)x

un(0)=uan, = 0,

(2.10) «„(0) = van,

(2.11)

wn(0) = wan,

VI) n C(S;

Un € CW(S;

V.),

(2.12)

V.) n C(S; H n ) , j

vn, wn 0 , uan —> ua

in

V 2,

the

van^»va

relations in

V and

wan-+wa

in

V

(2.14)

imply

un->u

in C(S; V),

vn^v

in C(8; H) and wn^w

in C(S; H),

(2.15)

where (u, v, w) is the solution of (2.1) —(2.4).

Remark 2.3. A procedure for numerical solving of (2.10)—(2.13) has been established in [8]. Some results of numerical calculations basing on this methods as well as a discussion of their physical significance may be found in [7]. 3. Proof of the basic existence theorem Before proving Theorem 1 we have still to make some preparations. According to the assumption of Theorem 1 there exists a sequence (( u an , van, wa„)j with (uan, va„, wan) € (C™(-n, n)) 3 such that uan->ua

in V p ,

van-+va

in H,

wan^wa

in H.

For this sequence and a sequence (e„) with s„ > 0 and e0 the problem (2.10)-(2.13). L e m m a 3.1. The problem with

unxx

€ CW(S;

vn € L\S-,

V 2n),

0 as n

(2.10) —(2.13) has a unique solution

Vn) a C(S; H„), vnt 6 C(S;

unt € GW(S; Vn),

(3.1)

(un, vn, w„) £ ( G { S ; F „ ) ) 3

V„) n C(S;

wnt £ C(S;

oo, we consider

Hn),

Hn) n L*(S;

V ).

86

H . GAJEWSKI

P r o o f . The existence of a unique solution (u n , vn, wn) of (2.10)—(2.13) with vn £ L2(S; Vi) n C{8; F„), vnt £ L2(8; Hn), wn £ C(S; V„), wnt £ C(S; H„) n L*(8; V„) has been proved in [8] (cf. [8], Satz 4.1, Bemerkung 4.1). The inclusions unxz £ CW(S; F„) n C(S; H„) and unt £ CW(S; V„) n C(S; Hn) can be shown by using wn £ C{8; F„) and wnt £ L2(S; F„) quite so as the corresponding result in [4] (Satz 5.2). Finally, vnl £ C(S; V„) follows from un £ C(S; V^), unt £ L2(S; F„) and wnt £ L2(S; F„) by applying known regularity results on linear parabolic equations (cf. [5], Satz 3). Lemma 3.2. It holds the estimate IWIctS;^) + IWIc(S;F>

IKIIc(S;FF) ^

(3.2)

C.

P r o o f . W e see from (2.10) that for each t £ 8

IM0IIÜ = ll«.(0)|i; + 2Re / (unt, un)n o

ds

t

= II«Jl 2 + 21m J o

(iunt

+

u

n x x

-

wnun,

un)n

= |[wa„||2 < c.

ds

(3.3)

Next one concludes from (2.10) and (2.12) that

IkJI«

=

+jwn

(2Re (u

J

\un\2dx +

n x t

ü

n x

) +

wn{\un\2)t

unxx

— wnun,

II2 + I M 2 ) (|K|

+

wnt

\un\2

+

+

e«

wnwnt

IKill®

+

vnvnl

+

env2nx)

dx

—n

=

—2Re

(iunt

+

u

n t

)

Integration with respect to t yields

n

+

(vnt

(IkJI 2 + I K . I P ) +

Jwn\unI2

M

dx

IIWX+

+

j K l ' dx

|w„|2

— envnx)x,

vn)„

= 0.

t

K | 2 dx + SnjKJI2

I Wan K„| 2 dx.

Now it holds n

(w>„

n

ll«.«Wlfi + J (IWOII 2 + IK(i)iln) + ft»n

= K.J8 + i

+

ds

(3.4)

Cauchy's problem for the one-dimensional Zakharov system

87

and by (1.2) and (3.3) n

JKl*

dx

si

I\un\\l \\un\\ln

=

K J

^ HttJIil

||W„[|„ + 2 i k j l )

—n 3

( — IIKJ + 2 \\u \\ ^ ( i K n l 1 + 2 i | Mnxn J ")

^ I K J 4 (( f ¿- + + 22 MM l)

+ +

li Wn-

Putting this in- (3.4), we obtain \\UnMfn

+ IMOli; + IKWII. +

/ ll®«li; 0

da

^ e-

(3.5)

In order to estimate the ffp„-norm |||m„|||„ we again use equation (2.10) and find by (3.1) and (3.5) t

Ill«.(0lll2 = lll«.(0)||i; + 2Re / o

{unt,T*un)nds

t

=

I I K J 112 + 2 I m J ((iunt

o

+

+ unxx

-

p2u„)n

wnun, t

(u«x> 2 P P x U n ) „ ) ds t

= ||K»III2 + 2 Im J o

(unx,

2 p p x u n ) n ds

0 t

^

Ci +C2f

\\\un\\\l

ds.

Now Gronwall's lemma says us that IWIcfS;^,) ^ c Because of Lemma 1.1 and Remark 1.3 from this and (3.5) the desired estimate (3.2) follows. Lemma 3.3. It

holds

lkJc(S;K*) ^ C. Proof. For arbitrary h £ Vn we get from (2.10) and (3.2) for every t 6 S I K ( i ) , h)\ = |(«.„(f)

-

wn(t)

u„(t),

^ IK,(Oil. IIM. + I W k IKIL »All. ^ c ll*lk, and hence the desired estimate.

88

H . GAJEWSKI

Lemma 3.4. The sequence (un) is in L2(S; H) relatively compact. P r o o f . The imbedding of Vp into H is compact (cf. [12]). According to Lemma 3.2 the sequence (w„) is in L 2 (S; Vp) bounded. Further we know from Lemma 3.3 that SUP | M U ' ( S ; F * „ )

n

SUP | | M

n

B (

|U»(S^

C.

Thus the assumptions of a compactness lemma of Groger [9] are satisfied which proves the assertion. Proof of Theorem 1. Because of Lemma 3.1—3.4 a subsequence ((«,-, vj} w,)) of solutions of (2.10) —(2.13) and elements u, v, w exist such that u ^ u

in

L2(S; V),

vi

uj->u

in

L2(S; H),

e^-^O

Here - > and (3.7) it follows

v

and

w]

in

w

in

L2(S;H),

L2(S; F ) .

(3.6) (3.7)

denote strong and weak convergence, respectively. From (3.2) and

ueL~(S-,Vp).

(3.8)

Taking into account (2.10), (3.6) and (3.7) we find for arbitrary test functions h € C^S; C0(-n, n)) with h(T) = 0 J (—(iu, ht) + (uxx — wu, hj) dt=

— J ((iu, ht) + (ux, hx) + (w, uh)) dt

s

s

= - lim f ((iUj, ht)n + (U jx , hx)„ + (Wj, Ujh)n) dt = lim ^iuaj, h(0))„ + J (iujt + ujxx — Wjiij, h)„ dtj = (m 0 , A(0)). Since the test functions lie densely in L2(S; F), we obtain using (3.8) that ut € L2(S; V*), u(0) = ua and that (2.1) holds as an equation in L2(S; F*). Further u £ Vp) and ut £ L2(S; V*) imply, as mentioned in the first section, u £ G(S; H) n CW(S; Vp). Next, using (2.11), (3.6) and (3.7), we get — / ((f, ht) + (w, hx) + (u, uhx)) dt s

= - lim J ({Vj, ht)n + {u>j, hx)„ + (uj, Ujhx)„) dt oo s = lim /(vaj, h(0))„ + J (vn + (Wj + \Uj\2 — env]x)x, h)n dt\ \

s

}

= {va,h( 0)). Hence by (3.6) and (3.8) it follows that vt £ L2{S; F*), v{0) = va and that (2.2) holds as an equation in L2(S; F*). Since v £ L2{S\H) a L2(S; F*), the inclusion vt £ L2(S; V*) implies v £ C(S\ F*) and by (3.2) v £ CW(S; H).

Cauchy's problem for the one-dimensional Zakharov system

89

The equation (2.3) and the relations w 6 C{S; V*) n OJS; H) and wt € L2{S; V*) may be derived like the corresponding statements concerning v. Then, wt € CU{S; V*) follows from (2.3) and v € GW(S; H). Finally, using w € CW(S; H) and u € Ca{8; V) n C(S; H), we obtain ut € CW{S; V*) and vt £ GW{S; V*) from (2.1) and (2.2). Remark 3.1. The existence proof just given becomes more simple if we replace the Cauchy problem (2.1) —(2.4) by the corresponding spatially periodic problem. In this case we can use the spaces Ha,b = L2(a, b) and Fa>t) = \u £ H^a, b) | u(a) = u(b)], where [a, 6] denotes the period. There is no reason for introducing spaces with weighted norms like Hp and Vp, because the embedding of V0ib into H0ib is compact. Corresponding remarks are true with respect to the proofs of the other theorems given in what follows. 4. Proof of the uniqueness result Proof ol Theorem 2. Evidently, it suffices to prove the inequation (2.6). For this purpose let («¿, w;), i = 1, 2, be solutions of (2.1) —(2.5) corresponding to the initial values (uai, va{, wai). We set u = ux — u2, v = — v2, w = Wi — w2 and denote by k , k u k 2 , ... constants depending on K||c + Kllc(S;F> +

c

Nlj'(S;V) + IKHc(S;n ^

(5-4)

-

Finally, from (2.10) —(2.13) and (5.4) we see that IKillc(S;Ji) + IMIi'(S;tf) + IKillctS;«) ^ C.

(5.5)

Remark 5.1. The expression (5.3) is closely related to the fifth conservation law of the nonlinear Schrodinger equation noted in [15]. Lemma 5.2.

Suppose

IW|c(s;F°)

ua

£

^

and

va,

wa

£

V.

Then

the

following

estimate

holds

C.

(5.6)

2

Proof. Since ua 6 we can assume that uan ua in V p. Using (1.1), (1.2), (3.2), (5.4) and (5.5), we find from (2.10), (2.12) i 2 lll«.«Wlli; = I I K J I I - 2 Re / ( « , „ 2 p p u ) dt - (w,(0 \u {t)\\ P% x

n t

n

n

o t +

\ua„\2,

(wan

p2)„

-

t \u„\2,p2)„ds

f (v„x


va

V,

in

Wan^-wa

in

V.

(5.9)

Let ((w„, v„, w j ) be the corresponding sequence of solutions of (2.1)—(2.4). From the proofs of Lemma 5.1 and Theorem 3 it is clear that nllL°°(S;K) +

IKill/,~(S;ff) + Ktllz;«(S;H) +

IKt||£oo(S;H) ^ C.

(5.10)

On account of (2.6), (5.9) and (5.10) there are functions u, v, w such that utt^u

in

C(S;V),

v„^v

in

C(S;H),

w„^w

in

C(8;H).

Now, as in the proof of Theorem 1, we see that (u, v, w) satisfies (2.1)—(2.4). The regularity properties (2.8) and (2.9) then follow by (2.1) —(2.4) and (5.10).

6. Proof of the approximation theorem Proof of Theorem 5. The existence of unique solutions (u„, vn, w„) of (2.10) —(2.13) satisfying estimates like (5.2) follows from Remark 3.1. It remains to prove the convergence result (2.15). B y assumption (2.14) we have ua 6 V2 and va, wa £ V. Thus we know from Theorem 4 that the solution ( u , v , w ) of (2.1) —(2.4) has the regularity properties (2.8) and (2.9). These imply \u{t, x)\ + |ux(t, £c)| +

\v(t, x)\ +

|w>(i, a:)l -9-0

as

x - > oo,

(6.1)

uniformly in t, and the existence of a sequence (y„) with yneC(8;C?«(-»,»)), V«t 6C{S-,C?(~n,

yn^u »)),

in

yttt^ut

in

L2(S; L2(S;

V2)

and

C(S;

H).

To abbreviate we set un = un — u,

vn = v„ — v,

wn = w„ — w,

y" — yn — u,

V),

1

(6.2)

94

H.

GAJEWSKI

Taking into account the regularity properties (2.8), (2.9), (2.13) and the estimate (5.2), we obtain from (2.1) and (2.10) t \\u«{t)\\l = ||«"(0)|£ - 2 Im / « - wnun + wu, u«)n ds o i

= l|wB(0)|£ - 2 Im f (K«"]" - K « " + w " , «")„) ^ o t = ll«"(0)||5 + / ([-«»*« - V» + uxu]"_n - (w„wn + uwn, u")„) ds (6.3) 0

where, because of (2.14), (5.2) and (6.1), ¿J -> 0 as n -> oo. Next we get KWII2 = Kit)

+ y»M\l < 2{\\y«(t)\\l + IKWII»)

= 2 {^ynM\l

+ 11*5(0)11; + / «

-

«? -

tf).

^

t

= dl + 2 R e j" (w„wn + WW", wf)„ rfs o = «5» + (»„(*), |w B «l 2 )» - K ( 0 ) , I«n(0)|2)„ + / (t>„, |m"|2)„ cb + 2 R e (u(t) wn{t), un{t))n - (w(0) wn(0), m"(0))„ - J (utwn - uvnx, un)„ ds = dl + («.(«), Wn{t)\% - (w.(0), |«"(0)| 2 ) b + f (vnx, \u«\*)n ds + 2 Re (u(t) u>n(t), «"( 0.

The function g can represent internal deformations or motions of the reference configuration. The force process / e H1 (S; U) corresponding to a given stress process a £ H^S; Y) is given by fit) = K*a(t)

\/t€S

(K* (i Hf(Y, TJ') denotes the adjoint operator of K). This follows easily from the principle of virtual work (cf. Moreau [9]). Therefore, if / € H^S; U') is a given process of "external" forces the condition of dynamic equilibrium is u"+K*a

= f.

(2)

In this equation we consider K* in the usual way as a mapping from L2(S; Y) into L2(S; U'). Similarly we shall deal later with other operators. For the sake of simplicity we have assumed that the system under consideration is of constant "density" 1. The condition of quasi-static equilibrium is K*a = f.

(2')

Let e be a strain process of the system and let a, a be corresponding processes of stress and internal parameter. Following Halphen-Nguyen [7] we assume that the following

99

Evolution equations in the theory of plasticity

constitutive relations hold: e = e + p,

e = A^a,

(p', —a') £ 8